Islamic historians began by translating Greek historians like Herodotus and Thucydides into Arabic. But soon there were historians of the Islamic Empire itself, who wrote in Arabic. One example is the huge history written by al-Tabari about 900 AD. Al-Tabari had done a lot of work, and his accounts of Islamic history are generally accurate and useful. Like Herodotus before him, if al-Tabari heard different accounts of a story, he wrote down both of them, leaving the reader to decide which was right.
Other historians wrote in Persian, using the Arabic alphabet. One of these was Ferdowsi, who wrote the Shahnameh. The Shahnameh was a history of the Persian kingdom, from the beginning of civilization down to his own time (about 1000 AD). Farouzi didn't know anything much about Cyrus the Great or Darius or Xerxes, but he did know about Alexander the Great, and about the later Sassanian kings, and the coming of Islam. Farouzi told many folk stories, too, like the story of Rustem and Sohrab and the story of Rudaba (who we call Rapunzel).
By the 1300s AD, the Islamic historian Ibn Khaldun, writing in Arabic in his home town of Tunis, was concerned not just to write a history of North Africa and the Berber people, but also to develop a scientific method for studying history, building on Aristotle and Plato to consider why empires rise and fall, rather than just reporting that they do.